Following a scoreless first half, Cologne forward Yuya Osako netted a 50-minute penalty for Japan after a handball at Toyota Stadium.
New Zealand striker Chris Wood scored an equalizer nine minutes later, triggering an all-out offensive by Japan.
On 87 minutes, Kurata scored his first goal for the Blue Samurai with a diving header, whipping up home fans in the heavy rain.
“I pushed the ball with all my heart,” Kurata said.
“I’m glad to contribute to the team with the winning goal,” he said.
It was Japan’s first friendly since they beat Australia 2-0 in August to qualify for their sixth successive World Cup.
“We won but it was not easy,” Japan’s head coach Vahid Halilhodzic said after the game.
“New Zealand displayed a brilliant defense,” said the Franco-Bosnian.
“We scored the first goal but could not control the game well and then they scored a goal,” he said.
But he praised his players for their collective offense toward the end of the game, which led to Kurata’s winner.
The defeat shadowed New Zealand, who will move to an intercontinental playoff next month.
Japan will take on Haiti in another friendly on Tuesday in Kanagawa, southwest of Tokyo.